# 4. Ambiguous Rings Based on a Heart Curve

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This Demonstration further explores ambiguous rings.

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Contributed by: Erik Mahieu (May 2018)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The parametric equation of with radius inclined at an angle from the vertical is given by:

,

with parameters and .

The parametric equation of over a heart curve in the - plane and an angular offset of from the axis [2] (the sixth heart curve is used in this Demonstration) is given by:

.

To find the intersection, we put the corresponding coordinates equal to get three equations in four unknowns: .

Eliminating , and by solving the equations gives the parametric curve of the intersection, with as the only parameter.

This composite curve (ring set) can be split into two rings. Therefore, the parameter range for , from to , is divided into sections using the cutoff angles and .

References

[1] E. Chicurel-Uziel, "Single Equation without Inequalities to Represent a Composite Curve," *Computer Aided Geometric Design*, 21(1), 2004 pp. 23–42. doi:10.1016/j.cagd.2003.07.011.

[2] E. W. Weisstein. "Heart Curve" from MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/HeartCurve.html (Wolfram *MathWorld*).

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